Cotton on a small branch (4)
I believe the answer is:
twig
I believe this is a double definition.
'cotton on' is the first definition.
('twig' can be a synonym of 'cotton on')
'a small branch' is the second definition.
(I've seen this before)
(Other definitions for twig that I've seen before include "Small branch - divining rod - realise" , "Understand; sprig" , "Suddenly catch" , "Finally understand" , "Suddenly realise" .)